The number of distinct real values of `lamda` for which the vectors `veca=lamda^(3)hati+hatk, vecb=hati-lamda^(3)hatj` and `vecc=hati+(2lamda-sin lamda)hatj-lamdahatk` are coplanar is

Options

A. 0
B. 1
C. 1
D. 3

Answer

B

Explanation

For `veca, vecb, vecc` to be coplanar, we must have
`[veca vecb vecc]=0`
`implies |(1,0,1),(1,-lamda^(3),0),(1,2lamda-sin lamda,-lamda)|=0
`implies lamda^(7)+lamda^(3)+2lamda=sin lamda`
This is true of `lamda=0`
For non-zero values of `lamda` it gives
`lamda^(6)+lamda^(2)+2=(sin lamda)/(lamda)`...................i
We know that `(sin x)/xlt1` for all `x!=0`. Therefore LHS of (i) is greater than 2 and RHS is less than 1. So i is that true for any non-zero `lamda`.
Hence, there is only one value of `lamda`